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Regularity of solutions and parameter estimation for SPDE's with space-time white noise

REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S
WITH SPACE-TIME WHITE NOISE
by
Igor Cialenco
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
May 2007
Copyright 2007 Igor Cialenco

In this work we discuss two problems related to stochastic partial differential equations (SPDEs): analytical properties of solutions and parameter estimation for SPDE's.; We address the problem of existence, uniqueness and regularity of solutions of some parabolic SPDE's driven by space-time white noise, either additive or multiplicative. The novelty in our study is the special form of the noise term which depends on a real parameter. We establish existence and uniqueness of weak solution in the scale of Sobolev spaces. Regularity properties of the solution are stated in terms of the real parameter involved in the noise term and spectral properties of the elliptic operator which generates the scale of Sobolev spaces.; We study the parameter estimation problem for some parabolic SPDEs with multiplicative stochastic part. Maximum Likelihood Estimators of the parameter, based on finite-dimensional approximation of the solution are found. Consistency, both in time and space variables, and asymptotic normality of these estimators are established. All theoretical results are followed by numerical simulations.

REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S
WITH SPACE-TIME WHITE NOISE
by
Igor Cialenco
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
May 2007
Copyright 2007 Igor Cialenco